How do you find the derivative of the function using the definition of derivative #g(t) = 7/sqrt(t)#?
1 Answer
The key step is to rationalize a numerator.
Explanation:
I'll assume that you are permitted to use the definition:
(There are other ways of expressing the definition of derivative, but this is a very common one.)
#= lim_(hrarr0)(7/sqrt(t+h)-7/sqrtt)/h#
#= lim_(hrarr0)(7sqrtt -7sqrt(t+h))/(sqrt(t+h)sqrtt)*1/h#
#= lim_(hrarr0)(7(sqrtt -sqrt(t+h)))/(hsqrt(t+h)sqrtt)#
Notice that, if we try to evaluate by substitution, we get the indeterminate form
The thing to try here (it will work) is to rationalize the numerator by using the conjugate of
That is: we will multiply by
We resume:
# =lim_(hrarr0) (7(t-(t+h)))/(hsqrt(t+h)sqrtt(sqrtt + sqrt(t+h))#
# =lim_(hrarr0) (-7cancel(h))/(cancel(h)sqrt(t+h)sqrtt(sqrtt + sqrt(t+h))#
Now we can evaluate the limit:
# = (-7)/(sqrttsqrtt(2sqrtt)) = (-7)/(t(2sqrtt)) = (-7)/(2tsqrtt)#
Note
It may be helpful to observe that in some sense we have traded the subtraction:
The subtraction goes to
In the process, we were able to eliminate the factor of
